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    <div class="post-body" itemprop="articleBody"><h1 id="哈希与字典树">哈希与字典树</h1>
<p>哈希（Hash）也叫散列，记录存储位置与关键字之间存在对应关系 <span
class="math inline">\(Loc(i)=H(key_{i})\)</span>，从而在查找数据时可以接近
<span class="math inline">\(O(1)\)</span>
的效率直接找到位置，而不需要顺序摸排或二分查找。</p>
<span id="more"></span>
<img src="/2025-03-12-10-%E5%93%88%E5%B8%8C%E4%B8%8E%E5%AD%97%E5%85%B8%E6%A0%91/hash_base.gif" class="" title="哈希基本原理">
<p>哈希函数自由度很高，能完成数据范围到哈希表范围的映射就可以，映射之后有概率产生冲突，通过一定手段解决冲突即可。</p>
<p>工程应用中，MD4、SHA-1、SHA-256、SHA-512 等都是哈希算法。</p>
<h3 id="链表法处理哈希冲突">链表法处理哈希冲突</h3>
<p>哈希表每个位置对应一条链表，所有散列值相同的元素都放到相同位置对应的链表中</p>
<img src="/2025-03-12-10-%E5%93%88%E5%B8%8C%E4%B8%8E%E5%AD%97%E5%85%B8%E6%A0%91/hash_link.gif" class="" title="哈希基本原理">
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line">list&lt;<span class="type">int</span>&gt; hl[maxn];</span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">Find</span><span class="params">(<span class="type">int</span> x)</span> </span>&#123;</span><br><span class="line">    <span class="type">int</span> ith = <span class="number">0</span>, hsNum = x % maxn;</span><br><span class="line">    list&lt;<span class="type">int</span>&gt;::iterator jt;</span><br><span class="line">    <span class="keyword">for</span>(jt = hl[hsNum].<span class="built_in">begin</span>(); jt != hl[hsNum].<span class="built_in">end</span>(); jt ++, ith ++) &#123;</span><br><span class="line">        <span class="keyword">if</span>(*jt == x) <span class="keyword">break</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">return</span> jt == hl[hsNum].<span class="built_in">end</span>() ? (hl[hsNum].<span class="built_in">push_front</span>(x), <span class="number">-1</span>) : ith + <span class="number">1</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h3 id="stl-的-map-与-unorderd_map">STL 的 map 与
<code>unorderd_map</code></h3>
<p>C++ 的 STL 利用红黑树封装了 <span
class="math inline">\(O(logn)\)</span> 的 <code>map</code>
映射，以及一套哈希方案封装的接近<span
class="math inline">\(O(1)\)</span>的 <code>unordered_map</code>
映射，用起来很方便。</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">std::unordered_map&lt;string, <span class="type">int</span>&gt; mp;</span><br><span class="line">mp[<span class="string">&quot;bob&quot;</span>] = <span class="number">5</span>;</span><br><span class="line">mp[<span class="string">&quot;alice&quot;</span>] = <span class="number">6</span>;</span><br></pre></td></tr></table></figure>
<p>不同版本的<code>C++</code>，<code>unordered_map</code>内部实现不尽相同，一些题目可能会专门Hack
<code>unordered_map</code> 导致其退化不再是 <span
class="math inline">\(O(1)\)</span>，所以比赛中如果理论复杂度用<code>map</code>没问题的话，还是推荐用<code>map</code>。</p>
<h2 id="字典树">字典树</h2>
<p>字典树是针对“串”类数据结构的哈希方案，比如对小写英文字符串做哈希，可以构建一个“26叉树”，将任意单词从左到右在树上进行分支查找，完成匹配。</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br></pre></td><td class="code"><pre><span class="line"><span class="type">const</span> <span class="type">int</span> maxn = <span class="number">1100</span>;</span><br><span class="line"><span class="keyword">struct</span> <span class="title class_">Node</span> &#123;</span><br><span class="line">    <span class="type">int</span> cnt;</span><br><span class="line">    <span class="type">int</span> nex[<span class="number">26</span>];</span><br><span class="line">&#125;;</span><br><span class="line">Node trie[maxn];</span><br><span class="line"><span class="type">int</span> tp;</span><br><span class="line"><span class="type">char</span> buf[<span class="number">11</span>];</span><br><span class="line"><span class="type">int</span> n;</span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">Insert</span><span class="params">(<span class="type">int</span> now, <span class="type">char</span> *b)</span> </span>&#123;</span><br><span class="line">    trie[now].cnt ++;</span><br><span class="line">    <span class="keyword">if</span>(!*b) <span class="keyword">return</span>;</span><br><span class="line">    <span class="keyword">if</span>(!trie[now].nex[*b - <span class="string">&#x27;a&#x27;</span>])</span><br><span class="line">    trie[now].nex[*b - <span class="string">&#x27;a&#x27;</span>] = tp ++;</span><br><span class="line">    <span class="built_in">Insert</span>(trie[now].nex[*b - <span class="string">&#x27;a&#x27;</span>], b + <span class="number">1</span>);</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">Search</span><span class="params">(<span class="type">int</span> now, <span class="type">char</span> *b)</span> </span>&#123;</span><br><span class="line">    <span class="comment">// 统计以输入字符串为前缀的单次个数</span></span><br><span class="line">    <span class="comment">// 对于其他任务（比如统计多少个单次是输入字符串的前缀）</span></span><br><span class="line">    <span class="comment">// 改变cnt的意义并维护对应值</span></span><br><span class="line">    <span class="type">int</span> nex = trie[now].nex[*b - <span class="string">&#x27;a&#x27;</span>];</span><br><span class="line">    <span class="keyword">if</span>(!*b) <span class="keyword">return</span> trie[now].cnt;</span><br><span class="line">    <span class="keyword">if</span>(!nex) <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">return</span> <span class="built_in">Search</span>(nex, b + <span class="number">1</span>);</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h3 id="思考题">思考题</h3>
<ol type="1">
<li>给n个单词，和1个长单词，求有多少个单词是这个长单词的前缀 —— 用 n
个单词建字典树，节点统计单词个数，长单词去匹配，路径上个数求和</li>
<li>给n个单词，和1个短单词，求这个短单词是多少个单词的前缀 —— 用 n
个单词建字典树，节点统计单词个数，短单词匹配成功的节点个数为答案</li>
<li>给两种语言单词对照表，输入任意单词，输出对照另一个语言的单词</li>
</ol>

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